The main purpose of this paper is to obtain the inference of parameters of heterogeneous population represented by finite mixture of two Pareto (MTP) distributions of the second kind. The constant-partially accelerated life tests are applied based on progressively type-II censored samples. The maximum likelihood estimates (MLEs) for the considered parameters are obtained by solving the likelihood equations of the model parameters numerically. The Bayes estimators are obtained by using Markov chain Monte Carlo algorithm under the balanced squared error loss function. Based on Monte Carlo simulation, Bayes estimators are compared with their corresponding maximum likelihood estimators. The two-sample prediction technique is considered to derive Bayesian prediction bounds for future order statistics based on progressively type-II censored informative samples obtained from constant-partially accelerated life testing models. The informative and future samples are assumed to be obtained from the same population. The coverage probabilities and the average interval lengths of the confidence intervals are computed via a Monte Carlo simulation to investigate the procedure of the prediction intervals. Analysis of a simulated data set has also been presented for illustrative purposes. Finally, comparisons are made between Bayesian and maximum likelihood estimators via a Monte Carlo simulation study.
References
[1]
Nelson, W. (1990) Accelerated Life Testing: Statistical Models, Data Analysis and Test Plans. John Wiley and Sons, New York. https://doi.org/10.1002/9780470316795
[2]
Bagdonavicius, V. and Nikulin, M. (2002) Accelerated Life Models: Modeling and Statistical Analysis. Chapman and Hall/CRC Press, Boca Raton, Florida.
[3]
Kim, C.M. and Bai, D.S. (2002) Analysis of Accelerated Life Test Data under Two Failure Modes. International Journal of Reliability, Quality and Safety Engineering, 9, 111-125. https://doi.org/10.1142/S0218539302000706
[4]
AL-Hussaini, E.K. and Abdel-Hamid, A.H. (2004) Bayesian Estimation of the Parameters, Reliability and Hazard Rate Functions of Mixtures under Accelerated Life Tests. Communications in Statistics-Simulation and Computation, 33, 963-982.
[5]
AL-Hussaini, E.K. and Abdel-Hamid, A.H. (2006) Accelerated Life Tests under Finite Mixture Models. Journal of Statistical Computation and Simulation, 76, 673-690. https://doi.org/10.1080/10629360500108087
[6]
Bai, D.S., Cha, M.S. and Chung, S.W. (1992) Optimum Simple Ramp-Tests for the Weibull Distribution and Type-I Censoring. IEEE Transactions on Reliability, 41, 407-413. https://doi.org/10.1109/24.159808
[7]
Wang, R. and Fei, H. (2004) Inference of Weibull Distribution for Tampered Failure Rate Model in Progressive Stress Accelerated Life Testing. Journal of Systems Science and Complexity, 17, 237-243.
[8]
Abdel-Hamid, A.H. and Al-Hussaini, E.K. (2007) Progressive Stress Accelerated Life Tests under Finite Mixture Models. Metrika, 66, 213-231.
https://doi.org/10.1007/s00184-006-0106-3
[9]
Miller, R. and Nelson, W. (1983) Optimum Simple Step-Stress Plans for Accelerated Life Testing. IEEE Transactions on Reliability, R-32, 59-65.
https://doi.org/10.1109/TR.1983.5221475
[10]
Bai, D.S., Kim, M.S. and Lee, S.H. (1989) Optimum Simple Step-Stress Accelerated Life Tests with Censoring. IEEE Transactions on Reliability, 38, 528-532.
https://doi.org/10.1109/24.46476
[11]
Gouno, E., Sen, A. and Balakrishnan, N. (2004) Optimal Step-Stress Test under Progressive Type-I Censoring. IEEE Transactions on Reliability, 53, 388-393.
https://doi.org/10.1109/TR.2004.833320
[12]
Fan, T.H., Wang, W.L. and Balakrishnan, N. (2008) Exponential Progressive Step-Stress Life-Testing with Link Function Based on Box-Cox Transformation. Journal of Statistical Planning and Inference, 138, 2340-2354.
[13]
Ma, H. and Meeker, W.Q. (2008) Optimum Step-Stress Accelerated Life Test Plans for Log-Location-Scale Distributions. Naval Research Logistics, 55, 551-562.
https://doi.org/10.1002/nav.20299
[14]
Nelson, W. (2008) Residuals and Their Analysis for Accelerated Life Tests with Step and Varying Stress. IEEE Transactions on Reliability, 57, 360-368.
https://doi.org/10.1109/TR.2008.920789
[15]
Wu, S.J. and Lin, Y.P. (2008) Optimal Step-Stress Test under Type I Progressive Group-Censoring with Random Removals. Journal of Statistical Planning and Inference, 138, 817-826.
[16]
Abdel-Hamid, A.H. and AL-Hussaini, E.K. (2008) Step Partially Accelerated Life Tests under Finite Mixture Models. Journal of Statistical Computation and Simulation, 78, 911-924. https://doi.org/10.1080/00949650701447084
[17]
Abdel-Hamid, A.H. (2009) Constant-Partially Accelerated Life Tests for Burr Type-XII Distribution with Progressive Type-II Censoring. Computational Statistics & Data Analysis, 53, 2511-2523.
[18]
Goel, P.K. (1971) Some Estimation Problems in the Study of Tampered Random Variables. Technical Rep. No. 50, Department of Statistics, Carnegie Mellon University, Pittsburgh, Pennsylvania.
[19]
DeGroot, M.H. and Goel, P.K. (1979) Bayesian Estimation and Optimal Designs in Partially Accelerated Life Testing. Naval Research Logistics, 26, 223-235.
https://doi.org/10.1002/nav.3800260204
[20]
Abdel-Ghani, M.M. (1998) Investigation of Some Lifetime Models under Partially Accelerated Life Tests. PhD Thesis, Department of Statistics, Faculty of Economics & Political Science, Cairo University, Cairo.
[21]
Ismail, A.A. (2004) The Test Design and Parameter Estimation of Pareto Lifetime Distribution under Partially Accelerated Life Tests. PhD Thesis, Department of Statistics, Faculty of Economics and Political Science, Cairo University, Cairo.
[22]
Ismail, A.A. (2009) Bayes Estimation of Gompertz Distribution Parameters and Acceleration Factor under Partially Accelerated Life Tests with Type-I Censoring. Journal of Statistical Computation and Simulation, 80, 1253-1264.
https://doi.org/10.1080/00949650903045058
Vidondo, B., Prairie, Y.T., Blanco, J.M. and Duarte, C.M. (1997) Some Aspects of the Analysis of Size Spectra in Aquatic Ecology. Limnology and Oceanography, 42, 184-192. https://doi.org/10.4319/lo.1997.42.1.0184
[25]
Childs, A., Balakrishnan, N. and Moshref, M. (2001) Order Statistics from Non-Identical Right Truncated Lomax Random Variables with Applications. Statistical Papers, 42, 187-206. https://doi.org/10.1007/s003620100050
[26]
Al-Awadhi, S.A. and Ghitany, M.E. (2001) Statistical Properties of Poisson-Lomax Distribution and Its Application to Repeated Accidents Data. Journal of Applied Statistical Science, 10, 365-372.
[27]
Howlader, H.A. and Hossain, A.M. (2002) Bayesian Survival Estimation of Pareto Distribution of the Second Kind Based on Failure Censored Data. Computational Statistics & Data Analysis, 38, 301-314.
[28]
Abd-Elfattah, A.M., Alaboud, F.M. and Alharby, A.H. (2007) On Sample Size Estimation for Lomax Distribution. Australian Journal of Basic and Applied Sciences, 1, 373-378.
[29]
Hassan, A.S. and Al-Ghamdi, A.S. (2009) Optimum Step Stress Accelerated Life Testing for Lomax Distribution. Journal of Applied Sciences Research, 5, 2153-2164.
[30]
Pearson, K. (1894) Contributions to the Mathematical Theory of Evolution. Philosophical Transactions of the Royal Society Series A, 185, 71-110.
https://doi.org/10.1098/rsta.1894.0003
[31]
Richardson, S. and Green, P.J. (1997) On Bayesian Analysis of Mixtures with an Unknown Number of Components (with Discussion). Journal of the Royal Statistical Society: Series B, 59, 731-792. https://doi.org/10.1111/1467-9868.00095
[32]
Ahmad, K.E., Jaheen, Z.F. and Mohammed, H.S. (2011) Finite Mixture of Burr Type XII Distribution and Its Reciprocal: Properties and Applications. Statistical Papers, 52, 835-845. https://doi.org/10.1007/s00362-009-0290-0
[33]
Ahmad, K.E., Jaheen, Z.F. and Mohammed, H.S. (2011) Bayesian Estimation under a Mixture of Burr Type XII Distribution and Its Reciprocal. Journal of Statistical Computation and Simulation, 81, 2121-2130.
https://doi.org/10.1080/00949655.2010.519703
[34]
Balakrishnan, N. and Aggarwala, R. (2000) Progressive Censoring: Theory, Methods and Applications. Birkhauser, Boston. https://doi.org/10.1007/978-1-4612-1334-5
[35]
Balakrishnan, N. (2007) Progressive Censoring Methodology: An Appraisal (with Discussions). Test, 16, 211-296. https://doi.org/10.1007/s11749-007-0061-y
[36]
AL-Hussaini, E.K. (1999) Predicting Observables from a General Class of Distributions. Journal of Statistical Planning and Inference, 79-91.
[37]
AL-Hussaini, E.K. and Ahmad, A.A. (2003) On Bayesian Interval Prediction of Future Records. Test, 12, 79-99. https://doi.org/10.1007/BF02595812
[38]
Jaheen, Z.F. (2003) Prediction of Progressive Censored Data from the Gompertz Model. Communications in Statistics-Simulation and Computation, 32, 663-676.
https://doi.org/10.1081/SAC-120017855
[39]
Ali Mousa, M.A.M. and AL-Sagheer, S.A. (2005) Bayesian Prediction for Progressively Type-II Censored Data from the Rayleigh Model. Communications in Statistics-Simulation and Computation, 34, 2353-2361.
https://doi.org/10.1080/03610920500313767
[40]
Xiong, C. and Milliken, G.A. (2002) Prediction for Exponential Lifetimes Based on Step-Stress Testing. Communications in Statistics-Simulation and Computation, 31, 539-556.
[41]
Kundu, D. and Howlader, H. (2010) Bayesian Inference and Prediction of the Inverse Weibull Distribution for Type-II Censored Data. Computational Statistics & Data Analysis, 54, 1547-1558.
[42]
Kundu, D. and Raqab, M.Z. (2012) Bayesian Inference and Prediction of Order Statistics for a Type-II Censored Weibull Distribution. Journal of Statistical Planning and Inference, 142, 41-47.
[43]
Zellner, A. (1994) Bayesian and Non-Bayesian Estimation Using Balanced Loss Functions. In: Berger, J.O. and Gupta, S.S., Eds., Statistical Decision Theory and Methods, Springer, New York.
[44]
Jozani, M.J., Marchand, E. and Parsian, A. (2006) Bayes Estimation under a General Class of Balanced Loss Functions. Rapport Derecherche 36, Departement de Mathematiques, Universite de Sherbrooke.
[45]
Upadhyay, S.K., Agrawal, R. and Smith, A.F.M. (1996) Bayesian Analysis of Inverse Gaussian Non-Linear Regression by Simulation. Sankhyā B, 58, 363-378.
[46]
Ahmadi, J., Jozani, M.J., Marchand, E. and Parsian, A. (2009) Bayes Estimation Based on k-Record Data from a General Class of Distributions under Balanced Type Loss Functions. Journal of Statistical Planning and Inference, 139, 1180-1189.
[47]
Ahmadi, J., Jozani, M.J., Marchand, E. and Parsian, A. (2009) Prediction of k-Records from a General Class of Distributions under Balanced Type Loss Functions. Metrika, 70, 19-33. https://doi.org/10.1007/s00184-008-0176-5
[48]
Upadhyay, S.K., Vasishta, N. and Smith, A.F.M. (2001) Bayes Inference in Life Testing and Reliability via Markov Chain Monte Carlo Simulation. Sankhyā A, 63, 15-40.
[49]
Press, S.J. (2003) Subjective and Objective Bayesian Statistics: Principles, Models and Applications. IEEE Transactions on Reliability, 57, 435-444.
[50]
Upadhyay, S.K. and Gupta, A. (2010) A Bayes Analysis of Modified Weibull Distribution via Markov Chain Monte Carlo Simulation. Journal of Statistical Computation and Simulation, 80, 241-254. https://doi.org/10.1080/00949650802600730
[51]
Metropolis, N., Rosenbluth, A.W., Rosenbluth, M.N., Teller, A.H. and Teller, E. (1953) Equations of State Calculations by Fast Computing Machines. Journal Chemical Physics, 21, 1087-1091. https://doi.org/10.1063/1.1699114
[52]
Robert, C.P. and Casella, G. (2004) Monte Carlo Statistical Methods. Springer, New York. https://doi.org/10.1007/978-1-4757-4145-2
[53]
Viveros and Balakrishnan (1994) Interval Estimation of Parameters of Life from Progressively Censored Data. Technimetrics, 36, 84-91.
[54]
Burkschat, M., Cramer, E. and Kamps, U. (2006) On Optimal Schemes in Progressive Censoring. Statistics & Probability Letters, 76, 1032-1036.
